Vector quantile regression (VQR) is an optimal transport (OT) problem subject to a mean-independence constraint that extends classical linear quantile regression to vector response variables. Motivated by computational considerations, prior work has considered entropic relaxation of VQR, but its fundamental structural and approximation properties are still much less understood than entropic OT. The goal of this paper is to address some of these gaps. First, we study duality theory for entropic VQR and establish strong duality and dual attainment for marginals with possibly unbounded supports. In addition, when all marginals are compactly supported, we show that dual potentials are real analytic. Second, building on our duality theory, when all marginals are Gaussian, we show that entropic VQR has a closed-form optimal solution, which is again Gaussian, and establish the precise approximation rate toward unregularized VQR.